Environmental Engineering Reference
In-Depth Information
where
c
s
1
and
c
s
2
are the concentrations of sorbed species, and
c
1
and
c
2
are the
concentrations of dissolved species. The exponents
n
1
and
n
2
are the electric
valence coefficients for the species 1 and 2.
K
is the characteristic equilibrium
constant for the exchange between two cations. For the competition between Ca
2+
and Na
+
the formula (
6.8
) delivers:
c
Ca;s
c
Na
c
Na;s
¼ K
(6.9)
p
c
Ca
Several equilibrium sorption approaches for cation competition are discussed by
Vulava et al. (
2000
). An alternative to (
6.8
) one may use the Gaines-Thomas
isotherm for exchangeable mass fractions on the porous medium (Engesgaard and
Christensen
1988
; Appelo et al.
1993
):
1
=n
1
1
=n
2
c
s
1
c
1
c
2
c
s
2
¼ K
(6.10)
In the formulation of mathematical analysis, given in (2.4), sorption can be
included by the introduction of the exchange terms. Considering advective and
diffusive fluxes, first order decay or degradation, neglecting additional sinks and
sources, the analytical formulation of the mass balances in both phases is:
@
@t
yðÞ¼ry
ðÞylc e
fs
j
(6.11)
@
@t
r
b
c
s
ð
Þ¼ r r
b
j
s
ð
Þr
b
l
s
c
s
e
sf
with concentrations
c
and
c
s
, porosity
and sorption exchange terms
e
fs
and
e
sf
. The
exchange term with subscript
fs
denotes the losses from the mobile to the immobile
phase and
sf
vice versa. The exchange terms have a positive sign for losses
in the first phase and a negative sign if the first phase gains due to exchange.
In comparison to the formulation, given in Chap. 3, porosity appears as coeffi-
cient in the storage, in the flux and in the decay terms. In these terms the additional
factor is relevant in order to take into account that storage, flux and decay occur in
the pore space only.
The second (
6.11
) describes the mass balance for the solid phase, the porous
medium. As the species concentration at the solid surface is usually given as a mass
fraction, the coefficient
y
r
b
has to appear in order to obtain the mass balance for the
r
b
[kg/m
3
] is the
bulk density
of the porous medium that is given by.
species.
r
b
¼
ð
y
Þr
s
1
(6.12)
where
r
s
is the density of the solid material without pores. On the right side of
the equation, the flux j
s
appears in order to denote fluxes in the solid phase.
